Magyar Tudományos Akadémia Számítástechnikai és Automatizálási Kutató Intézete

Computer and Automation Institute, Hungarian Academy of Sciences

PROBLEMS FROM THE WO R L D S U R R O U N D I N G P E R F E C T GRAPHS

A. GYÁRFÁS

Tanulmányok 177/ 1985 Studies 177/1985

1984-ben j e l e n t e k m e g :

155/ 1984 Deák, Hoffer, Mayer, Németh, Potecz, Prékopa, Straziczky: Termikus erőmüveken alapuló villamosenergiarendszerek rövidtávú, optimális, eromüvi menetrendjének meghatározása hálózati feltételek figyelembevételéve 1.

156 /1984 Radó Péter: Relációs adatbáziskezelő rendszerek összehasonlitó vizsgálata

157/1984 Ho Ngoc Luat: A geometriai programozás fejlődései és megoldási módszerei

158/ 1984 PROCEEDINGS of the 3rd International Meeting of Young Computer Scientists,Edited by: J. Demetrovics and J. Kelemen

159/ 1984 Bertók Péter: A system for monitorina the machining operation in automatic manufacturing systems

160/1984 Ratkó István: Válogatott számítástechnikai és matematikai módszerek orvosi alkalmazása

16 1/1984 Hannák László: Többértékü logikák szerkezetéről.

162/ 1984 Kocsis J. - Fetviszov V. : Rugalas autamatizált rendszerek: megbizhatóság és irányítási problémák

163/1984 Kalavszky Dezső: Meleghengermüvi villamos hurokemelő hajtás vizsgálata

164/1984 Knuth Előd: Specifikációs adatbázis modellek

165/1984 MTA SZTAKI Publikációk 1983. Szerkesztette: Petróczy Judit

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ABS TRAC T

A family G of graphs is called X-bound with binding function f if X(G' )

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C O N T E N T

0. Introduction1. x "bound and 0-bound families and their binding func

tions1.1. Basic concepts1.2. Some examples of X-bound and 0-bound families1.3. Algorithmic aspects of binding functions

2. Binding functions on families with one forbidden subgraph

3. Binding functions on families with an infinite set of forbidden subgraphs

4. Binding functions on families having a self-complementary set of forbidden subgraphs

5. Binding functions on union and intersection of graphs6. Complementary binding functions and the stability of

Perfect Graph Theorem

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0. INTRODU CTION

My aim is to introduce and propose a systematic study of x~bound (and 6-bound) families of graphs and their binding functions. These families are natural extensions of the world of perfect graphs. Recall that the family P of perfect graphs contains the graphs G which satisfy X(G')=

The examples in 1.2 (e.g. circular arc graphs, multiple interval graphs, box graphs, polyomino graphs, overlap graphs) show that the behaviour (or at least the known properties) of these families are quite different concerning their binding functions. Although these families are usually X-bound and 0-bound (the exception is the family of box graphs for more than two dimensions), in most cases the order of magnitude or linearity of their smallest binding function is not known.

The significance of binding functions from algorithmic point of view is discussed in 1.3. The idea is that families having "small" x-kinding functions (6-binding functions) are natural candidates for approximation algorithms with a "good" performance ratio for the coloring problem (clique cover problem). The smaller is a binding function of a family, the better performance ratio is to be expected from an approximation algorithm operating on the graphs of the family.

Perfect families of graphs are often characterized by a set of forbidden induced subgraphs. The family of P^-free graphs, Split graphs, Treshold graphs, Triangulated graphs, Meynel graphs are examples of such families. Analogous questions are discussed in sections 2,3 and 4 for X-bound families of graphs: which forbidden induced subgraphs make a family X-bound? Section 2 presents problems and results concerning the following conjecture: the family of graphs which does not contain a fixed forest as an induced subgraph is X-bound. In section 3 we discuss problems when the set of forbidden induced subgraphs is infinite. The Strong Perfect Graph Conjecture fits into this problem area. It is surprising

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that a much weaker conjecture, namely that the family of graphs without odd holes and their complements is X-bound, seems to be difficult. We should call this conjecture the Weakened Strong Perfect Graph Conjecture. In section 4 we consider the case when the set of forbidden subgraphs is closed under taking complementary graphs.

In section 5 we study the effect of taking union and intersection of graphs on binding functions. It is straightforward that the union of x-bound families is again a X- bound family. However, the intersection of two X-bound families (even the intersection of two perfect families) is not necessarily X-bound.

The situation of having the notion of x-bound and 0-bound families resembles the time B.P.G.T. (Before Perfect Graph Theorem) when two types of perfectness had to be defined. It is easv to construct families which are X- bound but not 0-bound although "natural" graph families are usually both X-bound and 0-bound. In section 6 we try to find analogons of the Perfect Graph Theorem for certain x-bound families of graphs. Let G^ denote the family of graphs 0-bound with 0-binding function f. If G^ is X-bound then the smallest X-binding function of G^ is called the complementary binding function of f. It turns out that the only self-complementary binding function is f(x)=x that is the Perfect Graph Theorem is stable in a certain sense. Only "small" binding functions may have complementary binding functions: if f has a complementary binding function then lirr inf f(x)/x=l. However, it remains an open problem even to prove that f(x)=x+l has a comple- menetary binding function.

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All results appearing here with proofs are unpublished elsewhere. They are expository in nature and mainly serve as background material and status information for the open problems. In fact, the main motivation of the author for writing this paper is his desire to see some of these 44 problems to be solved. I am indebted to my friend and collegue J.Lehel for several discussions which helped these ideas to take shape.

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1, X-BOUND AND 6- B O U N D FAMILIES AND THEIR B I N D I N G FUNCTIONS

1.1. Basic concepts. Let cj(G) and x( G ) denote the clique number and the chromatic number of a graph G, i.e. co(G) is the maximum number of pairwise adjacent vertices of G and X(G) is the minimum number k such that the vertices of G can be partitioned’ into k stable sets. A subset of vertices in a graph is called stable if it contains

̂pairwise non-adjacent vertices.A function f is a x-binding function for a family G

of graphs if

X-(G') < fU(G'))

holds for all induced subgraphs G' of GGG.Concerning the function f, we shall always assume

that f:N-HN where N denotes the set of positive integers, moreover f(l)=l, f(x)>x for all xGN.

A family G of graphs is x-bound if there exists a X-binding function for G.

The above definitions can be formulated for the complementary parameters of graphs. Let a(G) and 6(G) denote the stability number and the clique-cover number of a graph G, i.e. a(G) is the maximum number of vertices in a stable set of G and 6(G) is the minimum number k such that the vertices of G can be partitioned into k cliques.

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A function f is a 9-binding function for a family G of graphs if

6(G') < f(ct(G'))

holds for all induced subgraphs G' of GGG. A family G of graphs is. 9-bound if there exists a 9-binding function for G.

Since d)(G)=a(G) and X(G)=9(G) holds for any graph G by definition (where G denotes the complement of G), we observe:

f is a x-binding function for G if and only if f is a 9-binding function for G;G is X-bound if and only if G is 0-bound;

where G denotes the family (G:G€G).If a family G is x~bound then it has obviously a

smallest X-binding function defined by

f *(x) = max{X( G ' ) : G' G6£?,w(G' )=x} .

Similarly, a 0-bound family has a smallest 0-binding function. rDue to the assumptions on binding functions, the smallest binding function a family may have is the iden- titv function f(x)=x. The family of graphs with X-binding function f(x)=x is the important family of perfect graphs. The family of perfect graphs is denoted by P.The Perfect Graph Theorem of Lovász (C263) states that P=P which implies that P can be equivalently defined as the family of graphs with Q-binding function f(x)=x.

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The basic problems in our approach concerning a family G of. graphs are:

Is G a X-bound (or 0-bound) family?What is the order of magnitude of the smallest X-bin- ding (or 8-binding) function for G?Determine the smallest x~binding (or e-binding) function for G.Before looking at some examples of x-bound or 0-bound

families, have a glance at the outside world. Let be a graph such that w(Gi)=2 and X(G^)=i, for each integer i>2. The existence of G^ is well-known, see for examole Í291. Nov/ the family {G2f^3r...} is obviously not X-bound since it is impossible to define the value of a x-binding function f(x) for x=2. A more surprising example of a family which is not X-bound is provided by the intersection graphs of boxes in the three dimensional Euclidean space (see in 1.2.).

1.2. Some examples of X-bound and Q-bound families. Now have a look at some well-known families of graphs and their binding functions. We start with three classical subfamilies of P which we need frequently later.Interval graphs: the intersection graphs of closed intervals in a line.Triangulated graphs: the graphs contining no Ĉ. (a cycle of k vertices) for k>4 as an induced subgraph.Comparability graphs: the granhs G whose edges can be oriented transitively (ab, bcGE(G) implies ac6E(2)).

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The proof of the perfectness of the above families can be found in Cl63. Vie continue with some well-known non-perfect families of graphs defined as intersection graphs of geometrical figures. Proof techniques and results concerning their binding functions have been surveyed in C22].Circular arc graphs (see in [163, p.188): the intersection graphs of closed arcs of a circle. The family of circular arc graphs is 0-bound, its smallest 6-binding function is f(x)=x+l. The family is x-t>ound as well, the function f(x)=2x is a suitable x-binding function for x>2. Both of these statements follow immediately from the perfectness of interval graphs. It is easy to construct circular arc graphs for all k, sat is fvincr u(G^)=k, X( Gj,) = [3k/2J . A.Tucker conjectures (see [363 that X(G)< [3oj(G)/2j holds for all circular are graphs G. In our terminology, Tucker's conjecture says:Conjecture 1.1. The smallest X-binding function for the family of circular arc graphs is f(x)= |_3x/2Multiple (or t~) interval graphs: intersection graphs of sets which are the union of t closed intervals of a line. Irt the special case when t=l, we get interval graphs. These graphs were introduced in [173 and in C2U□„ The results of [213 imply that the family of t-interval graphs is 0-bound for all fixed t. The order of magnitude of the smallest 0-binding function is not known even for t=2 .Problem 1.2. Determine the order of magnitude of the smallest e-binding function for double interval graphs.In particular, is there a linear e-binding function for double interval graphs?

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It was proved in 1201 that the family of t-interval graphs is X-bound with a linear binding function 2t(x-l) for x>2.Box graphs (introduced in C333): intersection graphs of sets of boxes in the d dimensional Euclidean space. A box is a parallelopiped with sides parallel to the coordinate axes. For d=l we have the family of interval graphs.

It is easy to see that the family of d dimensionaljbox graphs is 6-bound with 6-binding function x (see proposition 5.5. later). The order of magnitude of the smallest 6-binding function is not known even for d=2.Problem 103. Determine the order of magnitude of the small- lest 6-binding function for two dimensional box graphs.

Concerning x-binding functions, it was proved byAsplund and Grunbaum (ill) that two dimensional box graphs

2are X-bound with an 0(x ) X“binding function. The order of magnitude of the smallest x~binding function is not known, its value at x-2 is 6 as proved in TilProblem 1.4. Determine the order of magnitude of the smallest X-binding function for two dimensional box graphs. In partucular, decide whether it is linear or not.

A surprising construction of Burlina (lUl) shows that the family of three dimensional boxes is not x-bound.Polyomino graphs. This subfamily of two dimensional box graphs received some attention in the last few years. A polyomino is a finite set of cells in the infinite planar square grid. With a polyomino P we may associate a

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hypergraph H(P) whose vertices are the cells of P and whose edges are the set of cells in maximal boxes contained in P. The intersection graph G(P) of H(P) may be called a polyomino graph. Obviously, G(P) is a subfamily of two dimensional boxes thus it is both e-bound and X-bound Answering a question of Berge et al. (C 31) , J.B.Shearer proved (II35 d ) that G(P) is perfect if P is simply connected. It would be interesting to see whether the family of polyomino graphs has linear binding functions, these questions are attributed to P.Erdos.Problem 1.5. Is there a linear 0-binding function for polyomino graphs?Problem 1.6. Is there a linear x_binding function for polyomino graphs?Overlap graphs (alias Circle Graphs, Stack Sorting Graphs see Cl6] p.242). These graphs are defined by closed inter vals df a line as follows: the vertices are the intervals and two vertices are joined by an edge if the corresponding intervals overlap, i.e. they are intesecting but neither contains the other. An equivalent definition is obtained by considering the intersection graphs of chords of a circle. Golumbic calls these graphs "not so perfect" (see [ 1 6 ], p.235). A measure of "non-perfectness" can be the order of magnitude of the smallest binding functions. It is easy to give an 0(x ) 0-binding function for the family of overlap graphs (see proposition 5.4 later). It is harder to prove that the family is X-bound, the smallest known x~binding function is exponential (see in C 20]).

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Problem 1.7. Is there a linear 0-binding function for the family of overlap graphs?Problem 1.8. Is there a linear ^-binding function for the family of overlap graphs?Intereeotion graphs of straight line segments in the plane. This family of graphs was introduced in C7]. The problem whether this family is x-bound (6-bound) arised during a conversation with P.Erdős. Denote this family by Gcto, just to have a temporary name for reference.

kDi-iO

Prob lem Problem

1.9.

1 . 1 0 .IsIs

(?Sls a X-bound family? GSLS a e-k°un

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case we have a polynomial approximation algorithm with perfomrance ratio at most f(a)(G))/w(G) which may or may not be satisfactory in a particular situation. A very favorable case occures when a family G has a linear X- binding function. Then the performance ratio of the algorithm is constant. The polynomial approximation algorithm can be useful if the coloring'problem is known to be NP-complete for the family G which is again a typical case. Similar reasoning shows the role of e-binding functions in approximation algorithms for the cligue cover problem. (The basic notions are used here as defined in Clltl).

To see some examples, consider the coloring problem for circular arc graphs. This problem is NP-complete (see in C153), on the other hand it is easy to give a polynomial approximation algorithm with performance ratio at most 2. The algorithm comes from the proof of the fact the 2x is a X“binding function for the family of circular arc graphs. If (_3x/2j were known to be a X-binding function (see conjecture 1.1.) then the proof would probably yield a polynomial approximation algorithm with performance ratio at most 3/2 .

The situation is similar if the coloring problem is considered for multiple interval graphs. The problem is NP-complete since the family of 2-interval graphs contains the family of circular arc graphs and the latter is NP- complete. The proof of the X“binding function 2t(x-l) for the family of t-intervals (x>2) provides a very simple polynomial approximative algorithm with performance ratio less than 2t (see in [203).

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The above reasoning might convince the reader about the importance of the following vaguely formulated problem.Problem 1.11. Find some applicable sufficient condition y/hich implies that a family has a linear X-binding function .

The existence of a linear binding function is an open problem for many x-koun(3 and/or 0-bound families. Problems1.2.-1.8. provide examples and we shall see others later.

Concerning potential applications, we note that the coloring problem of circular arc graphs and multiple interval graphs occures in scheduling problems (see C302, C2Í+3, Cl6D ) , applications of the coloring problem of overlap graphs are discused in Ll6D. The clique cover problem of polyomino graphs is motivated by problem of picture processing as noted in C3H.

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2, BINDING F U N C T I O N S ON F A M I L I E S WITH ONE F OR BIDD EN SUBGRAPH

Let H be a fixed graph and consider the family G(H) of graphs which does not contain H as an induced subgraph :

G(H) = {G:H£G}.

What choiches of H guarantee that G(H) is a x"bound family? Assume that H contains a cycle, say length k. Let be a graph of chromatic number i and of girth at least k+1. The existence of such graphs was proved by Erdos and Hajnal in [103. Clearly G^SGÍH) for i=l,2,... showing that G(H) is not x~bound. I Conjectured that G(H) is x-bound in all other cases, i.e. the following holds.Conjecture 2.1. (C193) G(F) is X-bound for every fixedforest F.

Let denote the star on n vertices and let R(p,q) be the Ramsey function that is the smallest m=m(p,q) such that all graphs of m vertices contain either a stable set of p vertices or a clique of q vertices. The following result shows that G(S ) is x-bound and its smallest X-binding function is close to the Ramsey function.Theorem 2.2. The family G(S ) is X-bound and its smallest X-binding function f* satisfies

R ( n-1, x+1) -1n-2 = f * ( x ) 4 R(n-1,x)

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for all fixed n, n>3.Proof. Let G be a graph on R(n-l,x+l)-l vertices such that G contains neither a stable set of n-1 vertices nor a clique of x+1 vertices. Clearly G6G(Sn) and X(G)>IV(G)I/n-2 which gives the lower bound for f*.

To see the upper bound, let GGG(Sn), gd(G)=x . We claim that the degree of any vertex of G is less than R(n-l,x). If some vertex PGV(G) has at least R(n-l,x) neighbors then the neighborhood of P contains either a stable set of n-1 vertices or a clique of x vertices. The first possibility contradicts to GGG(Sn) and the second contradicts to u)(G)=x and the claim follows. Therefore the chromatic number of G is at most R(n-l,x). □

Note that for n=3 the lower and upper bounds are the same showing that f*(x)=x, i.e. G(S^) is a perfect family. It is easy to see that G(S^) consists of graphs which can be written as the union of disjoint cliques.Problem 2.3. Improve the estimates of theorem 2.2. for the smallest X-binding function ofG(S^).

The next special case when conjecture 2.1. is solved occures if the underlying forest is a path.Theorem 2.4. Let P^ denote a path on n vertices, n>2.Then G(P ) is X-bound and fn(x)=(n-1)x ̂ is a suitable n 11X-binding function.Proof. Considering n^l fixed, we prove by induction on új(G ) . To launch the induction, note that the theorem trivially holds for graphs G with (G)=1. Suppose that

X“ 1(n-1) is a binding function for all G'€G(P ) suchnthat ü)( G, ) < t for some t>l.

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Let GGG(Pn) and o(G)=t+l. Assuming that x(G)>(n-l)t, we shall reach a contradiction by constructing a path (Ql,Q2,... ,Q ) induced in G. Technically we define nested vertex sets V( G )̂ V( G^ ) ...3V(G. ) and verticesQ16V(G1), Q26V(G2), . ..,Q GVCG.̂ ) for all i satisfying, l

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which allows us to choose a connected component H of G„t-1 asatisfying X(H)>(n-(i+1))(n-1) . Since is connected

by (i), there exists a vertex Qi+1eA such that V(H)U{Qi+ }̂ induces a connected subgraph which we choose as G^+-̂. It is easy to check that G^ ,G2 , . . . ,Gi+1 and Qj.' ̂ 2 ' * ' ’'^i + 1 satisfy the requirements (i)-(iii).

Assume that B-0. Now X ( G . )^X (G ,)fl which implies t-1 t-1 1 A(n-i)(n-l)

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Problem 2.7. What is the order of magnitude of f*(x)?Problem 2.8. Determine c=lim f*(2)/n. (It is easy to see

n-*-°°that 1/24041. )

Combining the ideas of the proofs of theorem 2.2. and theorem 2.4, it is possible to prove that G(B) is X-bound where B denotes a broom. A broom is a tree defined by identifying an endvertex of a path with the center of a star. The broom is the maximal forest for which conjecture 2.1 is known to be true, in the following sense: if F is a forest which is not an induced subgraph of a broom then conjecture 2.1. is open. In particular, the following three special cases of conjecture 2.1 are open problems.Problem 2.9. Prove that G ( ̂ >*— ) is X“bound.

Problem 2.10. Prove that G ( —— — Í—»— •) is x-bound.

Problem 2.11. Prove that G ^ J is X-bound.It seems hard to attack the following special case of

conjecture 2.1: a X~binding function f(x) for G( F) can be defined at x=2 if F is a forest. To settle this problem, it is clearly enough to consider the case when F is a tree since every forest is an induced subgraph of some tree. Thus we haveConjecture 2.12. Let T be a tree and let G be a triangle- free graph which does not contain T as an induced subgraph. Then X(G)

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Problem 2.13. Prove conjecture 2.12. for the tree above.In what follows, we consider problems concerning the

smallest X-binding functions of some special forests. The first example is mK2, the uni°n °f m disjoint edges. Note that mK2 is an induced subgraph of P3m_^ therefore GCmK^) is X-bound by theorem 2.4. Theorem 2.4 gives an exponential X-binding function for GCmK^). The methods used in C393 give better results.Theorem 2.14. (Wagon C393). The family G(mK2) has an 0(x2 m̂ ̂) X-binding function.Theorem 2.15. (Wagon C393). The function ( 2 ) is a X-binding function for G(2K2)Problem 2.16. What is the order of magnitude of the smallest X-binding function for G(2K2)?

Problem 2.16. was posed in L39H and arose again in connection with a problem of Erdős and El-Zahar (£93). Wagon notes in C39 3 that 3x/2 is a lower bound for the smallest X-binding function of G(2K2). A much better lower bound is

R(C,,K . 4' x+1

where R(C4fKx+]_) denotes the smallest k such that everygraph on k vertices contains either a clique of size x+1or the complement of the graph contains (a cycle onfour vertices). The above lower bound is non-linear be-1+0cause R(C^,K^_) is known to be at least t for some e>0 as proved by Chung in C53. Concerning particular values of the smallest X-binding function f* for G(2K2), it is

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easy to see that fí:(2) = 3. Erdős offered 20$ to decide whether f*(3)=4. The prize v/ent to Nagy and Szetmiklóssy who proved that f*(3)=4. (C 30□)

Now we turn our attention to the smallest x-binding function of G(F) where F is a forest of four vertices.The number of such forests is six and three of them (P^,S. and 2K_) have been discussed before. The smallest X-binding function of G( : : ) is asymptotically R(4,x+1) as the next proposition shows.Proposition 2.17. Let f*(x) be the smallest X-binding function for G(::). Then

R( 4 , x+1) -1 „ . R(4,x+1)+2R( 3,x+l) n5 I l x j 5 -L*

Proof. The lower bound is obvious. Let p be the maximum number of disjoint three-vertex stable sets in GGG(::).Let I V( G) I =3p+q , then q < R(3,x+1)-1 and

X(G) < p+q -lV ĜH +2q < R(4,x+1)-1+2^R( 3,x+l)-l> =

R( 4 , x+1) + 2R( 3 , x+1) □_ -L •

The smallest X-binding function of G ( < •) isasymptotically R(3,x+1).Theorem 2.18. Let f*(x) be the smallest x-binding function of G ( * ) . Then

R(3,x+1)-1 , R(3,x+l)+x-2— — -— =r- < f*(x) < ------

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The lower bound is obvious. The proof of the upper bound is based on the following lemma.Lemma 2.19. Assume that G6G ( •) and aCG)^. Let Sbe a maximal stable set of G, i.e. |s|=a(G). Then oj(G-S) = u> (G ) — 1.Proof.: Let S={s^,s2,...,} and vGV(G)-S. Since G6G ( *)v is adjacent with either exactly one vertex of S or with alLvertices of S. Therefore V(G)-S=V1UV2 where v6V1 is adjacent with exactly one vertex of S and vGV2 is adjacent with all vertices of S. Let W be a clique of V(G)-S. Assume that w^ ,w2SW rk V1, w^#w2 and w^s^SEÍG), w2s_.eE(G), i^j . Since |s|>3, we can chose s^GS such that kfi, k^j . Now {w1 ,w2 , s± , } (or (w-ĵ ,w9 , s ̂ , sk} ) induces < • in G, which contradicts to G6G «We conclude that all vertices of WAV^ are adjacent with the same vertex, say s^GS. Clarly s^ is adjacent with all vertices of WftV2 . Therefore any clique of V(G)-S can be completed to a larger clique by adding a suitable vertex of S. □Proof of theorem 2.18. The theorem is trivial if a(G)=l. Assume that a(G)=2 and let x^y^, x2^2''*''Xp^p a maximum matching of G. Let q=|v(G)|-2p, then x(G)q. Thus

X(G )

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Now we can proceed by indcution on w(G). The case a)(G)=1 is trivial. The inductive step follows from lemma 2.17 and from the fact that the Ramsey function R(x,3) is strictly increasing. Let a(G) 3 and let S be a stable set of size a (G) . The inductive hypothesis can be applied to G ' =G-S thus

X(G)

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3, B I N D I N G F UN CTIO NS ON F A M I L I E S WITH AN I N F INIT E SET OF F OR BIDD EN S UB GRAP HS

Let H be a set of graphs and let G(H) denote the family of graphs containing no graphs of H as induced subgraphs

G(H) = {G : H

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that is a x~bound family with x-binding functionf(x)=x. Surprisingly, it is not even known that G(H U H )o ois X-bound.Conjecture 3.2. (Weakened Strong Perfect Graph Conjecture.)The family G(H \JH ) is x~bound.-1 o o

The Strong Perfect Graph Conjecture gives a necessary and sufficient condition for perfectness in terms of forbidden subgraphs. To state similar conjectures for families having binding functions other than f(x)=x seems difficult. Consider, for example, the family of graphs with 6-binding function f(x)=x+l. Graphs of that family do not contain the (disjoint) union of G^ and G0 as an induced subgraph where Ĝ ,G^.HJJH^. The following proposition show that "critical" graphs can be much more complicated. Since its proof is based on case analysis, we state it without proof.Pro-position 3.3. Let G be the graph shown on figure 1.Then 6(G)=a(G)+2 and every induced proper subgraph G'CGsatisfies 6 ( G ' ) 2 where Hq={ C2m+i,c2m+3'* *-}*

A weaker version of conjecture 3.5 seems also interesting:

29

Figure 1

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Conjecture 3.6. The family G(C^) is x-k°unß, where = i C£ 'ci + \ ,ci + 2 ' ’ ' ’ * '

Note that G(C^) is the family of triangulated graphs which is perfect. However, for l>5, the conjecture is open.

Special cases of the Strong Perfect Graph Conjecture are known to be true. Some of these results say thatG(H) is perfect if H=H UH U{H} where H is a four-vertexo ograph. J.Lehel was curious about the four vertex graphsH for which the perfectness of G(H UH U{H}) is not knowno oThe Perfect Graph Theorem reduces the eleven cases tosix. The perfectness of G(H \jH u(H}) is known in the folo olowing cases:

H = ta (A.Tucker C 373)H = ízt (K . R.Parthásarathy,G.Ravindra C 32 3 )H = (K.R.Parthasarathy,G.Ravindra [313)H = A - (Consequence of Meyniel's theorem [283

and a direct proof follows from lemma 2.19)

H = n (Seinsche proved that G ([" ) is perfect [3t3)

It remains to solveConjecture 3. 7. (J.Lehel). The family G(H \JH UÍC.})-o o 4=G(H UiC4)) is perfect.

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L\, BINDING FUNCTIONS ON FAMILIES HAVING A SELF- COMPLEMENTARY SET OF FORBIDDEN SUBGRAPHS

A family G of graphs is self-complementary if G=G i.e. GGG if and only if G6G. A self-complementary family G is X~bound if and only if G is 0-bound. Moreover, if G is X- bound then the smallest X-binding function of G is the same as the smallest 0-binding function of G. Therefore we can speak about binding functions of G without referring to x or to 0. We mention two well known families of perfect self-complementary graphs.Permutation graphs (see in Cl6l): graphs G such that both G and G are comparability graphs.Split graphs (see in Cl6l): graphs G such that both G and G are triangulated graphs. Equivalently, split graphs are graphs whose vertices can be partitioned into a clique and a stable set.

Let H be a family of graphs. Obviously G{H) is self- complementary if and only if H is self-complementary. In what follows, we investigate binding functions of G{H) for self-complementary H. To see some perfect families first, note that G ( n ) is perfect ([3^]), G ( Q , J 1, O ) is perfect and coincides with the family of split graphs as proved by Földes and Hammer (C133 ) . A slightly more general result is in Z211 (theorem 3). The family o ( a , 11, n ) is a subfamily of both previous families, thus it is perfect. The family contains the so called treshold graphs (see in Cl63)

Concerning the existence of binding functions, the main open problem is a special case of conjecture 2.1.

32

Conjecture 4.1. The family G(F,F) has a binding function for every fixed forest F.

It seems useful to look at some special cases of conjecture 4.1. A straightforward attempt is to settle the following weaker versions of problems 2.9-2.11.Problem 4.2. Prove conjecture 4.1 for F = P > - <Problem 4.3. Prove conjecture 4.1 for F = 3 — •Problem 4.4. Prove conjecture 4.1 for F =

Another problem is to determine or estimate the smallest binding function of G(F,F) when G(F) is known to be x~bound. The rest of the section is devoted to problems and results of this kind.Problem 4.5. Estimate the smallest binding function ofG(S ,S ) n n ( is a star on n vertices.)n

Concerning special cases of problem 4.5, note that the case n=3 is trivial since G(S^,S^) contains only cliques and their complements.The case n=4 is settled by the following theorem (cf. theorem 2.2).Theorem 4.6. The smallest binding function of G(S^,S^)(the claw and co-claw free graphs) is f(x)=Proof. Let G be a non-perfect member of G(S^,S^). The result of Parthasarathy and Ravindra (E 31d ) implies that G contains an induced odd cycle or its complement. By symmetry we may assume that C2k+i“ v̂i,v2'* * *'v2k+l^ is an an- duced subgraph of G for some k>2.

We claim that any vertex X6V( G ) -V( 02^-^ ) is adjacent to all or to no vertices of c2k+i*

33

To prove the claim, assume that x is adjacent to v^.If x is not adjacent to an3 thenVi-4,x> induces S4> In all cases we got a contradiction.Therefore x is adjacent to all vertices of C„. ,, and the2k+lclaim is proved.

Let V(G)-V(C2^+3)=AONA where A(NA) denotes the set ofvertices adjacent (non-adjacent) to C either A or NA is empty.

2k+l We claim that

Assume that aGA, beNA and abGE(G). Let v±v.^E(g ), now {a,b,v±,Vj} induces S4. Similarly, if ab$E(G) then we choose i and j such that v^v ^GECg ) and now {a,b,vifv.} induces S4. Thus the claim is true.

The theorem follows by induction on the number of vertices of GSG(S4,S4). The inductive step goes as follows.

Let GGG(S4,S4). If G is perfect then X(G)=o>(G) <

< j. Otherwise G=C2k+i'JA or G“C2]-' + lG,NA as was Prove

34

X(G) = X(A) + 3 < 3(u(G.)-2)2 J « - L ^ j

In the second case we use the inductive hypothesis for NA X (G) = X(NA) and u(G)=w(NA) thus

We proved that f(x) = is a binding function forG(S.,S„). To see that it is the smallest one, let G be 4 4 mdefined as follows. Consider K and remove the edqes ofm| mI -g-1 vertex disjoint C,-. Now it is easy to see that Gm6G(S4,S4) for all m, moreover oj(G5k )=2k, j(/(G5k) = 3k,“(G5k+l‘)=2k+1, x(G5k+l) = 3k+1* DProblem 4.7. Estimate the smallest binding function of G(P ,§n) (cf. theorem 2.4 and problem 2.6).Problem 4.8. What is the order of magnitude of the smallest binding function for G(P^,P^)? (Cf. problem 2.7.)Problem 4.9. What is the order of magnitude of the smallest binding function for G(mK2,mK2)? (Cf. theorem 2.14).

The case m=2 in problem 4.9 is settled by the following theorem.Theorem 4.10. The smallest binding function Gk2K2, 2K2) is f(x)=x+l (Cf. problem 2.16).Proof.: Let G6G(2K2,2K2) and let S be a stable set of G such that |s|=a(G). Assume that x,ySV(G)-S, xy^E(G). The definition of S and.2K2 qír G imply that r( x)AS and r(y)flS are non-empty sets and one contains the other, say r ( x)A SCT (y )/"iS . (r(p) denotes the set of neighbors of pGV(G).) Now 2K2 ^ G implies |r(x)AS|=l.

35

Let K^={x:x€V(G)-S, |r(x)OS|>l), then is clique inG by the argument above. We proceed to show that V(G)-(.SuK^) is again a clique of G. Assume that p,qGV(G)-(S K^) and pq^E(G). By definition, | r (p )n S | = | r (q)n S | = 1. However, r (p )0 S=r ( q )r»S contradicts the maximality of S and r(p)/iS £ #r(q)/3S contradicts the assumption 2K^ G.

We have shown that the deletion of a stable set S of G results in a perfect graph (the complement of a bipartite graph). Thus X(G>4X(G-S ) + l=u>(G-S ) + l 4 (jj(G) + 1, showing that f(x)=x+l is a binding function for G(2K^,2K^). To see that f(x)=x+l is the smallest binding function, it is enongh to consider complete graphs from which the edges of a are deleted. □

The proof of theorem 4.10 givesCorollary 4.11. If GGG(2K2,2K2 ) then V(G) can be partitioned into two cliques and a stable set. By symmetry, V(G) can be also partitioned into two stable sets and a clique.

Using lemma 2.19, it is easy to proveTheorem 4.12. Let F denote the forest < • . ThenG(F,F) contains complete multipartite graphs and their complements, moreover the graph C,-.

Using the result of Parthasrathy and Ravindra (C32□ ) which proves the Strong Perfect Graph Conjecture for c< 123 ) (or equivalenly for £?(•—••♦)) it is easy to deriveTheorem 4.13. Let F denote the forest •— •• • •Then the non-perfect members of G(F,F) are

1. The graph of figure 2 and its non-perfect subgraphs .

36

2. A clique K whose vetices are adjacent with two consecutive vertices of a C,-.

3. The complements of the graphs defined in 1. and 2.

Figure

Putting together the previous two theorems, we have the following corollary.Corollary 4.14. Let F denote either < * or •— • • •Then the smallest binding function of G(F,F) is

f (x)3 if x=2 x if x>2.

Before finishing this section, note that the smallest binding function of G(F,F) was found for four-vertex forests F with one exception. The exceptional case occures when F=K^, i.e. F is a stable set of four vertices. The

37

family G(K^,K^) is very excentric since it is finite (like G(K ,K ) in general for fixed m). Its smallest binding function f*(x) is determined by the values f*(2) and f*(3). It is easy to deduce f*(2)=3 from the facts that R(3,4)=9 and that a graph G with w(G)=2, X(G)>4 satisfies |V(G)|>9 (in fact, |V(G>|>11 is true as proved by Chvatal in C6□). It is possible to determine f*(3) without brute force?

38

5. BINDING FUNCTIONS ON UNION AND INTERSECTION OF GRAPHSk k

For graphs G, ,G0, . . . ,G,, the graphs U G. and O G.-L Z. K • 1 1 . -q A.1=1 1=1are defined usually as follows.

V(UGi) =UV(G±) E(UGi) -UB(Gi )

V(AGi) =nv(G±) E(AGi) -AE(Gi).

If G^,G^/•••r̂ k are families of graphs then their union is the family (CAP :G^GG^} and their intersection is the family (HG^ :G^€Gh } . By definition, 0 G^ is a X-bound family if and only if UG. is a 0-bound family. This fact combined with x(G-^oG2) i x(G-^)X(G2) gives the following obvious observation.Proposition 5.1.

a) If G^ , C?2 , . . . , G^ are x -bound families with bindingfunctions f,, f f , then UG. is a X-bound fami ^ 1 2' k lly and n f. is a suitable x~binding function,

i=1 1b) If G^,G2,...,G^ are 0-bound families with binding

functions f, ,f f. then C\G. is a 0-bound fami1 2 k lly and nf^ is a suitable 0--binding function. □

Proposition 5.1, triviales it is, sometimes can be conveniently applied to prove the existence of binding functions.Corollary 5.2. Let P denote the family of all perfect graphs. The union (intersection) of k copies of P is X- bound (0-bound) with binding function x . □

39

Problem 5.3. What is the smallest X-binding function for PUP?

Proposition 5.4. The family of overlap graphs is 6-bound2with 6-binding function x .

Proof. Let G-̂ denote the family of co-interval graphs,let G~ denote the family of interval inclusion graphs.

̂ 2 Since G^ and G ̂ are perfect families, x is a x-bindingfunction for G^LlG2 by corollary 5.2. The family of overlap graphs is a subfamily of G-̂ u g2. □Proposition 5.5. The family of d-dimensional box graphs is 6-bound with 6-binding function x .Proof. The family in question is the intersection of d families of interval graphs and we can apply corollary

It is tempting to think that C\ G. is x-bound providedi=l 1

that G^ is X-bound for i=l,2,...,k. However, this is notthe case. It may happen that GjT\G2 is not x~bound althoughG-ĵ and G ̂are perfect families. A suprising contructionof Burling (C^3) gives three dimensional box graphs forall positive integers n such that w(Bn)=2, X(Bn)=n. Theresult shows that Inlnl is not x-bound, where I denotesthe family of interval graphs. The analysis of Burling'sconstruction shows however that inJ is not X-bound,where J is the family of "crossing graphs" of boxes inthe plane. The vertices of crossing graphs are boxes inthe plane and two vertices are adjacent if and only ifthe corresponding boxes cross each other. It is immediateto check that J is a subfamily of the family of comparabi-

2lity graphs. Note that 1/1 I is X-bound with an 0(x ) X- binding function as proved by Asplund and Grünbaum (C13).

40

Therefore the results in cUl and in Cl] implyTheorem 5.6. Let I,C denote the family of interval graphs and comparability graphs, respectively. Then

a) I AI is x-bound b ) I A I A 1 is not X-boundc) I 0 C is not X-bound.Perhaps part a) holds in a stronger form.

Problem 5.7. Let t denote the family of triangulated graphs. Is t A t X-bound? In particular, is t a I X-bound?

Since the graphs of t can be represented as subtrees of a tree (see in Cl6n), problem 5.7 can be wieved as a geometrical problem.

The following result shows a pleasant property of comparability graphs. *Proposition 5.8. Let £ denote the family of Comparabilitygraphs. The intersection of k copies of C is X-bound and 2k-1X is a suitable x~binding function.

Proof.: Let G-̂ ,G2 / • • • > G^S £. and assign a transitive orientation to the edges of G. for áll i (l^i^k). Assume that

k 1xytHL( AG.). The edge xy is oriented according to its

i=l 1orientation in , moreover we assign a type to it asfollows. The type of xy is a 0-1 sequence of length k-1.For all j, l^j^k-l the j-th element of the sequence is 0if xy is oriented in Ĝ from x to y and it is 1 otherwise.It is immediate to check that the edges of a fixed type

kof n G. define a transitively oriented graph. The number

i=1 k-1 kof possible types is at most 2 which implies that A G.i=l 1

41

k-1can be written as the union of at most 2 comparability- graphs. Now the proposition follows from corollary 5.2. qProblem 5.9. Estimate the smallest ^-binding function oft At.

A subfamily of perfect graphs, the permutation graph* occure in many applications. Permutation graphs can be defined as graphs G such that both G and G are comparability graphs. Corollary 5.2 and proposition 5.8 givesProposition 5.10. Let k be fixed and consider the family G of graphs obtained by at most k applications of intersections and unions from permutation graphs. Then G is X-bound and 0-bound. □

Now we want to determine the smallest 0-binding function of families obtained as the union of k bipartite graphs. Observe that this family contains exactly the graphs of chromatic number at most 2 . Therefore we are interested in finding the smallest 0-binding function for the family Gm of at most m-chromatic graphs.Proposition 5.11. Letfunction for G . Then m

a) f*(x)m

f*(x)m denote the smallest 9-binding

b) f*(x) > ^ x for x>x =x (m).m = 2 o o

Proof. It is trivial to cover the vertex set of an most m-chromatic graph G by the vertices of at most bipartite graphs, ,B2,...,Bs. Now

s s0(G) < Z 0(B . ) = E a(B . )

42

The lower bound is pointed out by Erdős, remarking that for n-in and for arbitrary m, there is a graph G=G(n,m) on kn vertices satisfying a(G)=n, |x

Qb) f'i-(x) > -̂x if x is divisible by 5.

Proof. First we prove a. We may assume that GSG^ is 3- chromatic. Let A^,A2,A3 be the color classes of G in a good 3 coloring of V(G). Let G12,G13,G23' be the sub~ graphs of G induced by A ^ A 2, A A 3, A^A^, respectively, Since G^j is a bipartite graph, 6(G^j)=a(G^j) which shows that VtG^j) can be covered by at most a(G) cliques (vertices or edges) of G^ for l^icj^.

We may assume that the clique cover of V(G^) covers all vertices of V(G^) exactly once. The cliques in thecovers of V(G12)' V(G33), V(G23) form a clique cover of G with at most 3a(G) elements and all vertices of G are covered exactly twice by these cliques. This cover can be partitioned into components where the cliques (edges and vertices) of each component are either the edges and the two endvices of a path (allowing two identical vertices as a degenerate case) or the edges of a cycle of length divisible by 3. It is easy to check that the vertices of a component of m cliques can be covered by at most 5m/9 cliques. These cliques are edges and vertices except for a component which forms a triangle, in this case the triangle is used instead of three edges. Therefore we get a

43

5 5a(G)clique cover of V(G) with at most 3ot(G)’g- = —̂ - cliques.The lower bound b) was quessed by Erdős who devised to

find a graph G with |v(G)|=15, a(G)=5, X(G)=3, w(G)=2. Really, such G exists as a subgraph of a 17-vertex graph H containing neither triangles nor six independent vertices (see H in C253). The graphs containing disjoint copies

8 xof G form a family with 0-binding function -g- for the cases when x is divisible by 5. □Problem 5.14. Let f*(x) be the smallest binding function

̂ 8 of G Determine lim f*(x)/x. (it is at least and' at5 x °̂°most by proposition 5.12.)

44

6. COMPLEMENTARY BINDING FUNCTIONS. THE STABILITY OF THE PERFECT GRAPH THEOREM

We say that a binding function f has a complementary binding function if the family G ̂of graph« with 0-binding function f is x_bound. The smallest ^-binding function of G^ is called the complementary binding function of f. Note that 0 and X can change role» in the definitions. We are interested in the following general problem.Problem 6.1. Which binding functions have complementary binding functions and what are their complementary binding functions?

Using the notion of complementary binding function, the Perfect Graph Theorem says that the function f(x)=x is a self-complementary binding function. (The converse statement is also true, see theorem 6.6 later.)

One feels that only "small" functions may have complementary binding functions. This is really the case as the next theorem shows.Theorem 6.2. If f(x) has a complementary binding function then lim inf f(x)/x=1.Proof. To prove the theorem, it is enough to show that f (x) = (l+e)x has no complementary binding function if e is a real number satisfying 0

45

̂afáf ̂< 2+e f°r induced subgraphs G £. (2)£Note that (2) implies that is a triangle-free graph.

Therefore (1) implies that the family G£= { ,G^,.•.} is not x“hound. We are going to prove that G^ is a 6-bound family with 0-binding function f (x).

Let G be an induced subgraph of G^. We have to prove that 0 (G) < (1+e ) a(G ). Since G is triangle-free, 0(G)=|V(G)| v(G) where v(G) is the cardinality of a maximum matching in G. We can express v(G) by the Tutte-Berge formula (see in C383 and in I 2D) as follows:

v (G ) min _|V(G)J + |a | -q(H)AcV(G)

(3)

where H denotes the subgraph induced by V(G)-A in G and a(H) denotes the number of odd components of H. Using (3) and 6(G) = |V(G)|-v(G), we can rewrite 0 ( G ) Consider the partition of {l,2,...,m} into I^,I2, d e f i n e d as follows:

ieiĵ if H.l is bipartite and |V(Hi)|iei2 if H . 1 is bipartite and |V(Hi)|í g i3 if H.1 is not bipartite.

is even;(5 )is odd;

46

We claim thatIV(H . ) I

a (Hi) > --2 ----- if ieilf

|V(H )I+1a( ) > ---- ^2--- if iei2, (6)

IV(H. ) 1+1 a(Hi} > 2( 1 + e ) if iei

The first two inequalities are obvious. To prove thethird one, let C„ . , be a minimal odd cycle of H. for^t+i 12t+lsome iGI^. Using (2) for c2t+l' we 9et C2t+1 ̂ >_ 2+ei.e. t>— which impliese

|V(H.) > 21+1 > — +1.= e (7)

Observing that (7) is equivalent with

1V(H±) I I V ( H± ) I+12+e ^ 2(l+e)~

|V(H ) Iand a(H^) > 2+---- by (2), we get the third inequalityof (6).

Now we use (6 ) to estimate a(G). Clearly m

a(G)>L a(H. ) = Z a(H.) + E a(H.)+ Z a(H.)> i=i 1 iei1 1 isi 2 1 iei3 1

JV(H) | + |I9UI3! V(H)+cKH)> ---- zm-,— t---- > -ött-;— r- since |V(H. ) I is even for= 2(l+e) = 2(1+e ) 1 i 1iSI^ by (5). Thus we proved (4) and the theorem follows. □

47 -

Theorem 6.2 gives a necessary condition for the existence of complementary binding functions. Concerning sufficient conditions, the main open problem is the following.Conjecture 6.3. The function f(x)=x+c has complementary binding function for any fixed positive integer c.

Conjecture 6.3 is open even in the case c=l. Probably this case already contains all the difficulties. An evidence supporting conjecture 6.3 is the following result.Proposition 6.4. If G denotes the family of graphs with 6-binding function f(x)=x+c then, for all GGG, to(G) = 2 implies x ^G)4 6 c+2.Proof. Assume that GSG, a)(G)=2. Clearly, I 4 0(G)4 a(G)+c which implies

a (G ) > 1VtG)I~2c (8)

Let C-ĵ be an odd cycle of minimum length in G, let C2 be an odd cycle of minimum length in the subgraph induced by V(G)-V(C.) in G, etc. We continue to defineT m _C1 ,C2, ' • ’ ,Cm untü the subgraph induced by V(G)- U VC^)in G does not contain odd cycles. Applying (8) for the

msubgraph C induced by U V(c.) in G, we get

i=l 1m

a ( C) < I a(C i=l

from which m^2c follows. A good coloring of V(G) can be defined by coloring V(C) with 3m colors and using two additional colors for the bipartite graph induced by V(G)-V(C). Therefore X(G)£3m+2

48

By a deep result of Folkman (C123) which answers a conjecture of Erdős and Hajnal, condition (8) implies x(G)t then f(x) has a complementary binding function. D

It does not seem to be a trivial problem to determine the complementary binding functions of any function different from f(x)=x. Perhaps the simplest problem of this type isProb lem 6.5. Let f be the binding function defined as

f (x)x if x^2 3 if x=2

What is the complementary binding function of f? Perhaps ! 3x I is the truth.

The following result shows that the Perfect Graph Theorem is stable in a certain sense.Theorem 6.6. If f(x) is a self-complementary binding function then f(x)=x for all positive integers.Proof. Assume that f is self-complementary.Case 1. Assume that f(2)=2. If f(x)fx for some x6N then we can choose k6N such that k>3, f(k)>k and f(x)*=x for x

49

be a X-binding function for iC2k+l^' f is not self-complementary. The contradiction shows f(x)=x for all x6N.• “'I3k-1 for some k.Case 2. Assume that f(2)>2 and f(k) <Consider the graph Ĝ. whose complement is j^j disjoint Cc and, for odd k, an additional isolated vertex. Now fis a 0-binding function for (Gk ) (a(Gk )=2, 0(Gk )=3) butfails to be a X"t)inding function for (G^) ((D (Gk ) =k,

Case 3. f(k) for all k6N. In this case theorem 6.2 implies that f(x) has no complementary binding function, again a contradiction. G

A generalization of the Perfect Graph Theorem (proved also by Lovász in C273) says that a graph G is perfect if

a (G *’ ) *w(G' ) > IV (G ' ) I

holds for all induced subgraph G' of G. The first step in searching analogous properties would be to settleProblem 6.7. Let G be the family of graphs G satisfying

a (G ' ) • aj (G ' ) > I V(G' ) |-1

for all induced subgraphs G' of G. Is it true that G is a X-bound (or, equivalently, 6-bound) family? If yes, what is the smallest binding function for G?

*

50

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